If we have Einstein-Hilbert Lagrangian so:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \mathcal L = (-g)^{1/2}R [/tex] for R Ricci scalar (¿?) then my question perhaps mentioned before is if we can split the metric into:

[tex] g_{ab}= Ndt^{2}+g_{ij} dx^{i} dx^{j} [/tex] N=N(t) 'lapse function'

then i would like to know if somehow you can split the Lagrangian into:

[tex] \mathcal L = (-g)^{1/2}R=N^{1/2}(g^{3})^{1/2}g^{00}R_{00}+N^{1/2}(g^{3})^{1/2}g^{ij}R_{ij} [/tex]

Following Einstein equations then [tex] R_{00}=T_{00}= \rho [/tex] 'energy density equation' and

[tex] \iiint (g^{3})^{1/2}R^{(3)}d^{3}x = 2T [/tex]

[tex] \iint (g^{(3)})^{1/2}R_{00}=\mathcal H = E [/tex]

Is some kind of Kinetic energy so the Einstein-Hilbert Lagrangian takes the form:

[tex] \mathcal L =\int_{a}^{b} dt(T-V) [/tex] a Kinetic plus potential term.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A bit introduction to foliation in GR and some question.

**Physics Forums | Science Articles, Homework Help, Discussion**